Bit Error Rate (BER) Equations
In ASK, noise directly affects the signal amplitude, making it the most vulnerable since the data is carried in amplitude changes. In FSK, data is represented by frequency variations, and because noise typically impacts amplitude more than frequency, FSK is more robust than ASK. In PSK, data is encoded in the signal phase, and BPSK specifically uses 180-degree phase shifts, creating the greatest separation between signal points and therefore achieving the lowest bit error rate (BER) for the same power level. BER formulas for ASK, FSK, and PSK modulation schemes.
ASK
FSK
PSK
erfc / Q-function (Click here)
The Mathematical Foundation
The Bit Error Rate is fundamentally derived from the Conditional Probability of Error, where we integrate the Gaussian Noise PDF over the decision region:
In digital communications, the argument $x$ inside the $Q(x)$ function represents the ratio of the signal's "safety margin" to the noise magnitude. In an AWGN channel, the noise standard deviation is defined as: $\sigma = \sqrt{N_0 / 2}$
BPSK
Distance to boundary is $\sqrt{E_b}$
$x = \sqrt{\frac{2E_b}{N_0}}$
BFSK
Distance to boundary is $\frac{\sqrt{E_b}}{\sqrt{2}}$
$x = \sqrt{\frac{E_b}{N_0}}$
BASK
Distance to boundary is $\frac{\sqrt{E_b}}{2}$
$x = \sqrt{\frac{E_b}{2N_0}}$
Why the 0.5 factor?
In 0.5 * erfc(x), the 0.5 represents the 50% probability of transmitting a '0' vs a '1' in a random bitstream.
Eb/N0 vs SNR
While SNR is the power ratio, Eb/N0 is the energy per bit, allowing us to compare schemes with different bandwidths fairly.
Theoretical BER vs SNR for Amplitude Shift Keying (ASK)
The theoretical Bit Error Rate (BER) for binary ASK depends on how binary bits are mapped to signal amplitudes. For typical cases:
If bits are mapped to 1 and -1, the BER is:
BER = Q(√(2 × SNR))
If bits are mapped to 0 and 1, the BER becomes:
BER = Q(√(SNR / 2))
Where:
- Q(x) is the Q-function:
Q(x) = 0.5 × erfc(x / √2) - SNR: Signal-to-Noise Ratio
- N₀: Noise Power Spectral Density
Understanding the Q-Function and BER for ASK
- Bit '0' transmits noise only
- Bit '1' transmits signal (1 + noise)
- Receiver decision threshold is 0.5
BER is given by:
Pb = Q(0.5 / σ), where σ = √(N₀ / 2)
Using SNR = (0.5)² / N₀, we get:
BER = Q(√(SNR / 2))
Theoretical BER vs SNR for Frequency Shift Keying (FSK)
For binary FSK, the theoretical BER is:
BER = Q(√(SNR))
The Q-function is defined as:
Q(x) = 0.5 × erfc(x / √2)
Similarities Between ASK and FSK
- Both BERs decrease as SNR increases
- Both use the Q-function for analytical BER calculation
- FSK generally performs better under noisy conditions
MATLAB Code for Theoretical BER vs SNR
Binary ASK (BASK)
% The code is written by SalimWireless.Com
clc;
clear all;
close all;
SNRdB = 0:20;
SNR = 10.^(SNRdB/10);
BER_th = (1/2) * erfc(0.5 * sqrt(SNR));
semilogy(SNRdB, BER_th, '-rh', 'linewidth', 2.5);
grid on;
title('Theoretical Bit Error Rate vs. SNR for Binary ASK Modulation');
xlabel('SNR (dB)');
ylabel('BER');
legend('Theoretical');
axis([0 20 1e-5 1]);
Binary FSK (BFSK)
% The code is written by SalimWireless.Com
clc;
clear;
close all;
SNRdB = 0:1:10;
SNR = 10.^(SNRdB/10);
BER_th = (1/2) * erfc(sqrt(SNR / 2));
disp('SNR (dB) Theoretical BER');
disp([SNRdB', BER_th']);
figure;
semilogy(SNRdB, BER_th, '-kh', 'LineWidth', 2);
xlabel('SNR (dB)');
ylabel('Bit Error Rate (BER)');
title('Theoretical BER vs SNR for BFSK');
grid on;
BER vs SNR Simulation
Comparison of ASK, FSK, and PSK Performance
| Feature | ASK (OOK) | BFSK | BPSK |
|---|---|---|---|
| Power Efficiency | Low | Medium | High |
| Bandwidth Efficiency | High | Low | High |
| Noise Immunity | Poor (Sensitive to Amp) | Good | Excellent |
| Best Used In | Fiber Optics, RFID | Caller ID, Paging | Deep Space, Satellite |
Key Terms Explained
- Eb/N0 (Energy per Bit to Noise Power Density):
- The normalized SNR measure, also known as the "SNR per bit." It is the most important metric for comparing different modulation schemes regardless of bandwidth. [Read More]
- Q-Function:
- The Q-function represents the tail probability of the standard normal distribution. In BER terms, it tells us the probability that noise will exceed the decision boundary. Read more
- Coherent vs. Non-Coherent Detection:
- Coherent detection (used in the formulas above) requires the receiver to be in phase-sync with the transmitter. Non-coherent detection is easier to build but results in a higher BER.